Preprint typeset in JHEP style - HYPER VERSION DESY 07-141 CP Violation and Neutrino Masses and Mixings from Quark Mass Hierarchies 2007 Wilfried Buchmuller Oct HewtscAes ^e&tronen-SyncArotron .DRST, TfamZmry, Germany 15 Laura Covi HewtscAes ^le&tronen-SyncArotron Ramtury, Germany David Emmanuel-Costa [hep-ph] GPTP, Reyartamento de RVca, /.strata Snyehor Teenzco, h;.s6on, Portuyal Soren Wiesenfeldt Reyarfmenf o/PAy.s;c.s, GmrerfVy 0/ TZZmots at Gr6ana-GAamya?yn, ABSTRACT: Wc study the connection between quark and lepton mass matrices in a supersymmetric SO(IO) GUT model in six dimensions, eompaetified on an orbifold. The physical quarks and leptons are mixtures of brane and bulk states. This leads to a characteristic pattern of mass matrices and high-cncrgy CP violating phases. The hierarchy of up and down quark masses determines the CKM matrix and most arXiv:0709.4650v2 charged lepton and neutrino masses and mixings. The small hierarchy of neutrino masses is a consequence of the mismatch of the up and down quark mass hicrar- ehics. The effective CP violating phases in the quark sector, neutrino oscillations and Icptogcncsis arc unrelated. In the neutrino sector wc can accomodate naturally sin #23 ~ 1, sin #13 <0.1 and mi < m2 ~ 1/Amj^ < m3 ~ i/Arn^. K eywords : CP violation, Field Theories in Higher Dimensions, GUT, Neutrino Physics. Contents 1. Introduction 1 2. SO(IO) Unification in six dimensions 2 3. CP violation in the quark sector 8 4. CP violation in the leptonic sector 11 4.1 Seesaw Mechanism and Effective Mass Matrix 12 4.2 Neutrinoless Double Beta Decay (Ov^) 15 4.3 CP Violation in Neutrino Oscillations 16 4.4 Leptogenesis 18 5. Conclusions 20 A. Mass matrices 22 A.l Down Quarks and Charged Leptons 23 A.2 Neutrinos 26 B. CP Violation and Weak Basis Invariants 35 1. Introduction Grand unified theories (GUTs) appear to be the most promising framework [1,2] to address the still challenging question of quark and lepton masses and mixings. During the past years new results from neutrino physics have shed new light on this problem, and the large differences between the mass hierarchies and mixing angles of quarks, charged leptons and neutrinos impose strong constraints on unified extensions of the Standard Model (SM) [3,4]. Massive neutrinos are most easily incorporated in theories with right-handed neutrinos, which leads to SO(IO) as preferred GUT gauge group [5,6]. Higher-dimensional theories offer new possibilities to describe gauge symmetry breaking, the notorious doublet-triplet splitting and also fermion masses. A simple and elegant scheme is provided by orbifold compactifications which have recently been considered for GUT models in five and six dimensions [7-12]. In this paper we analyse in detail the connection between quark and lepton mass matrices in the 1 six-dimensional (6D) GUT model suggested in [13], for which also proton decay [14], supersymmetry breaking [15] and gauge coupling unification [16] have been studied. An alternative SO (10) model in five and six dimensions has previously been studied in [17]. For a recent discussion of CP violation in a 5D orbifold GUT model, see [18]. An important ingredient of orbifold GUTs is the presence of split bulk multiplets whose mixings with complete GUT multiplets, localised at the fixed points, can sig ­ nificantly modify ordinary GUT mass relations. This extends the known mechanism of mixing with vectorlike multiplets [19-21]. Such models have a large mixing of left-handed leptons and right-handed down quarks, while small mixings of the left­ handed down quarks. In this way large mixings in the leptonic charged current are naturally reconciled with small CKM mixings in the quark current. Our model of quark and lepton masses and mixings relates different orders of magnitude whereas factors O(1) remain undetermined. Hence, we can only discuss qualitative features of quark and lepton mass matrices. Recently, orbifold compact- ihcations of the heterotic string have been constructed which can account for the standard model in four dimensions and which have a six-dimensional GUT structure as intermediate step very similar to familiar orbifold GUT models [22-24]. In such models the currently unknown O(1) factors are in principle calculable, which would then allow for quantitative predictions. The goal of the present paper is twofold: As a typical example, we first study the model [13] in more detail and explicitly compute the mass eigenstates, masses and mixing angles. Second, we investigate the question of CP violation, both in the quark and lepton sector and possible connections between the two. In previous studies, CP violation has mostly been neglected assuming that, barring fortunate cancellations, the phases and mixings are practically independent. Nevertheless this question and the Savour structure are strongly interconnected, and we will see that a specific pattern of mass matrices can give a distinct signature also in the CP violation invariants. This paper is organised as follows: In Section 2 we describe the 6D orbifold GUT model and the diagonalisation of the mass matrices defining the low energy SM fermions. In Section 3 we discuss the CP violation in the quark sector, whereas Section 4 is devoted to the CP violation in the leptonic sector. Conclusions are given in Section 5. Two appendices provide details to the computation of the mass eigenstates and CP violation in extensions of the SM. 2. SO(IO) Unification in six dimensions We study an SO (10) GUT model in 6D with N =1 supersymmetry compactihed on the orbifold T2/(Z2 x Z^ x ) [11,12]. The theo^ has four fixed points, Oi, Ops, Oca and Og, located at the four corners of a 'pillow' corresponding to the two compact dimensions (cf. Fig. 1). The extended supersymmetry is broken at all fixed 2 Ofl [Gfl] Oi [80(10)] Figure 1: The three SO(IO) subgroups at the eorresponding fixed points (braucs) of the orbifold x x points; in addition, the gauge group SO(IO) is broken to its three subgroups Gps = SU(4) x SU(2) x SU(2); Gee = SU(5) x U(%; and hipped SU(5), Gn = SU(5)' x U(l)% at Ops, Oee and On, respeetively. The interseetion of all these GUT groups yields the standard model group with an additional U(l) factor, G Sm' = SU(3) x SU(2) x U(l)y x U(l)^„, as unbroken gauge symmetry below the compactihcation scale. The held content of the theory is strongly constrained by imposing the cancclla- tion of irreducible bulk and brane anomalies 1251. The model proposed in Ref. 1131 contains three spinors 0^(16), % = 1... 3, as branc hclds as well as six vectorial hclds Hj(10), j = 1.. .6, and two pairs of spinors, $(16) + $ c(16) and 0(16) + 0C(16) as bulk hypermultiplets. The massless zero modes #($) and #"($") acquire vacuum expectation values (vevs), r'Af = (Y) = (A^), breaking B — A and thus G^p to Gsw- The breaking scale is close to the compactihcation scale so that n^/M* ^ 10^ GeV, where M* is the cutoff of the 6D theory. At the weak scale, the doublets Hd(Hi) and HU(H2) acquire vevs, vi = (Hd) and v2 = (Hu), breaking the electroweak symmetry. The three sequential 16-plets are located on the three branes where SO(IO) is broken to its three GUT subgroups; in particular, we place 0i at Ooo, 02 at and "03 at Ops. The parities of 0, and arc chosen such that their zero modes, have the quantum numbers of a lepton doublet and antidoublet as well as anti-down and down-quark singlets, respectively. Both A(0) and fR(0^) arc SU(2)^ doublets. Together these zero modes act as a fourth vectorial generation of down quarks and leptons. The three 'families' 0^ arc separated by distances large compared to the cutoff scale M*. Hence, they can only have diagonal Yukawa couplings with the bulk Higgs hclds; direct mixings arc exponentially suppressed. The branc hclds, however, can mix with the bulk zero modes for which we expect no suppression. These mixings 3 take place only among left-handed leptons and right-handed down quarks, leading to a characteristic pattern of mass matrices [13,14]. The mass terms assume the characteristic form, W + d*m^ + e^m^ m^ (2.2) where latin indices only span 1, 2, 3, while greak indices include the forth generation states. The up quark and Majorana neutrino mass matrices, mu and mN, are diagonal 3 x 3 matrices, 0 0 0 \ /h 0 \ 0 ) ’ 11 M, 2 2 mN = 0 0 (2.3a) m hU V '^22 M, 0 hggV 2/ 0 0 Since vC is pa^ of an SU(2)L doublet, it cannot couple to the other SM singlets in ^ via the B — L breaking held. Furthermore, there is no other coupling giving it a direct Majorana mass. The Dirac mass matrices of down quarks, charged leptons and neutrinos, md, me and mD, respectively, are 4 x 4 matrices instead, due to the mixing with the bulk held zero modes, o o 0 hggfi o md (2.3b) 0 0 hggfi \flVN fsVN y ^ h <f1vi 0 0 h14vi^ ^hf1v2 0 0 hf4v2^ e 0 h22vi 0 h24vi 0 hU2V2 0 h24v2 m (2.3c) 0 0 h^vi h^4Vi , 0 0 hUgV 2 h|4v2 ^ Ml M2 M3 M4 / ^ Ml M2 Mg M4 y up to corrections O(vN /M;2). The diagonal elements satisfy four GUT relations which correspond only to the local unbroken groups, i.e., SU(5), hipped SU(5) and Pati-Salam subgroups of SO(IO). The hypothesis of a universal strength of Yukawa couplings at each hxpoint leads to the identihcation of the diagonal and off-diagonal elements of mu/ tan0, md, me, and mD/ tan0, where tan0 = v2/v1, up to coeffi­ cients of order one.